Question 1194166: Of a random sample of 172 minority small business owners, 118 said that personal savings was the most important source of initial financing. Test the null hypothesis that personal savings is the most important source of initial financing for 75% of minority small business owners against the alternative that the population percentage is less than 75%.
Answer by math_tutor2020(3817)   (Show Source):
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p = population proportion of minority business owners who say that personal savings was the most important source of initial financing.
Hypotheses:
H0: p = 0.75
H1: p < 0.75
This is a left-tailed test because of the inequality sign in the alternative hypothesis.
n = 172 = sample size
x = 118 = number of successes
phat = sample proportion (which estimates the population proportion p)
phat = x/n
phat = 118/172
phat = 0.686047 approximately
About 68.6% of the sample say that savings is the most important source.
Based on this alone, it appears that the alternative hypothesis (H1) is correct. However, this may be due to random chance and why we need to proceed with the hypothesis test.
Compute the standard error (SE)
SE = sqrt(p*(1-p)/n)
SE = sqrt(0.75*(1-0.75)/172)
SE = 0.033017 approximately
Compute the test statistic
z = (phat - p)/SE
z = (0.686047 - 0.75)/0.033017
z = -1.93697186297968
z = -1.94
Standard practice is to round z scores to two decimal places. Though be sure to follow your teachers instructions if s/he says otherwise.
Now use a table such as this one
https://www.ztable.net/
Highlight the row that starts with -1.9
Highlight the column that starts with 0.04 at the top
This row and column combo overlap at the value 0.02680 which is approximate
This table is saying P(Z < -1.94) = 0.02680 approximately.
This is the p value as it represents the probability of getting a z score of -1.94 or smaller.
You did not state what the alpha level is, but if we assume the default of 0.05, then we'd reject the null hypothesis.
Rule: If the p-value is smaller than alpha, then reject the null.
Because we rejected the null, we conclude that the alternative hypothesis is indeed correct. It appears that p < 0.75 is true.
In other words, it appears that less than 75% of the business owners think personal savings is the most important part of initial financing.
Keep in mind that if alpha was something like 0.01, then the p-value is no longer smaller than it and we'd fail to reject the null (aka "accept" the null).
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